Surface kinetics and generation of different terms in a conservative growth equation

A method based on the kinetics of adatoms on a growing surface under epitaxial growth at low temperature in (1+1) dimensions is proposed to obtain a closed form of the local growth equation. It can be generalized to any growth problem where surface morphology is governed by adatom diffusion. The method can be easily extended to higher dimensions. The kinetic processes contributing to various terms in the growth equation are identified from the analysis of in-plane and downward hops. In particular, processes corresponding to the term that breaks h-->-h symmetry and the curvature dependent term are discussed. Effects of these terms on the stable to unstable transition in (1+1) dimension are analyzed. In (2+1) dimensions, it is shown that an additional asymmetric term is generated due to the in-plane curvature associated with mound-like structures. This term is independent of any diffusion barrier differences between in-plane and out-of-plane migration. It is shown that terms generated in the presence of downward hops are the relevant terms in a growth equation. A growth equation in closed form is obtained for various growth models introduced to capture most of the processes in experimental molecular beam epitaxial growth. The effect of dissociation is also considered and is seen to have a stabilizing effect on growth. It is shown that for uphill current the growth equation approach fails to describe the growth since a given single equation does not apply over the entire substrate.